How to use AI to depth map video?
I think that the tasks you are referring to are the Simultaneous localization and mapping (SLAM) and, in particular, the Structure from Motion (SfM).

These methods are usually based on geometrical constraints and do not employ neural networks, but there exist some recent methods that make use of CNNs (such as this one).

Structure from Motion algorithms are a fundamental component of the Google ARCore (former Project Tango) and the Apple ARKit.
Unfortunately, these kits usually provide an interface to put elements in a specific position, but they don't provide the access to the 3d reconstruction of the scene.

If you'd like to play with SfM algorithms, I suggest you to start from this repository, but you can find a lot of other valuable tutorials online.